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Chapter 11 - Rolling, Chapter 11 - Rolling, Torque, and Angular Torque, and Angular Momentum Momentum

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Chapter 11 - Rolling, Chapter 11 - Rolling, Torque, and Angular Torque, and Angular Momentum Momentum Chapter 11 - Rolling, Chapter 11 - Rolling, Torque, and Angular Torque, and Angular Momentum Momentum Rolling Angular Momentum Conservation of Angular Momentum “The way to catch a knuckleball is to wait until it stops rolling and then pick it up.” -Bob Uecker David J. Starling Penn State Hazleton PHYS 211 Each velocity vector on the wheel is a sum of ~vcom and the rotation. Chapter 11 - Rolling, Rolling Torque, and Angular Momentum Rolling Rolling motion is a combination of pure rotation Angular Momentum Conservation of Angular and pure translation. Momentum Each velocity vector on the wheel is a sum of ~vcom and the rotation. Chapter 11 - Rolling, Rolling Torque, and Angular Momentum Rolling Rolling motion is a combination of pure rotation Angular Momentum Conservation of Angular and pure translation. Momentum Each velocity vector on the wheel is a sum of ~vcom and the rotation. Chapter 11 - Rolling, Rolling Torque, and Angular Momentum Rolling Rolling motion is a combination of pure rotation Angular Momentum Conservation of Angular and pure translation. Momentum Chapter 11 - Rolling, Rolling Torque, and Angular Momentum Rolling Rolling motion is a combination of pure rotation Angular Momentum Conservation of Angular and pure translation. Momentum Each velocity vector on the wheel is a sum of ~vcom and the rotation. Note: therefore, acom = Rα. Chapter 11 - Rolling, Rolling Torque, and Angular Momentum The relationship between the angular and linear Rolling Angular Momentum velocities is fixed if the wheel does not slip. Conservation of Angular Momentum vcom = R! Chapter 11 - Rolling, Rolling Torque, and Angular Momentum The relationship between the angular and linear Rolling Angular Momentum velocities is fixed if the wheel does not slip. Conservation of Angular Momentum vcom = R! Note: therefore, acom = Rα. In this case, there is a static force of friction at the pivot that does no work. Chapter 11 - Rolling, Rolling Torque, and Angular Momentum Rolling If the rolling object does not slip, then the point in Angular Momentum Conservation of Angular contact with the floor is momentarily stationary. Momentum Chapter 11 - Rolling, Rolling Torque, and Angular Momentum Rolling If the rolling object does not slip, then the point in Angular Momentum Conservation of Angular contact with the floor is momentarily stationary. Momentum In this case, there is a static force of friction at the pivot that does no work. 1 = I + MR2 !2 2 com 1 1 = I !2 + M(R!)2 2 com 2 1 1 = I !2 + Mv2 2 com 2 K = KR + KT Chapter 11 - Rolling, Rolling Torque, and Angular Momentum Rolling Angular Momentum The kinetic energy of the wheel rotating about Conservation of Angular point P is Momentum 1 K = I !2 2 p 1 1 = I !2 + M(R!)2 2 com 2 1 1 = I !2 + Mv2 2 com 2 K = KR + KT Chapter 11 - Rolling, Rolling Torque, and Angular Momentum Rolling Angular Momentum The kinetic energy of the wheel rotating about Conservation of Angular point P is Momentum 1 K = I !2 2 p 1 = I + MR2 !2 2 com 1 1 = I !2 + Mv2 2 com 2 K = KR + KT Chapter 11 - Rolling, Rolling Torque, and Angular Momentum Rolling Angular Momentum The kinetic energy of the wheel rotating about Conservation of Angular point P is Momentum 1 K = I !2 2 p 1 = I + MR2 !2 2 com 1 1 = I !2 + M(R!)2 2 com 2 K = KR + KT Chapter 11 - Rolling, Rolling Torque, and Angular Momentum Rolling Angular Momentum The kinetic energy of the wheel rotating about Conservation of Angular point P is Momentum 1 K = I !2 2 p 1 = I + MR2 !2 2 com 1 1 = I !2 + M(R!)2 2 com 2 1 1 = I !2 + Mv2 2 com 2 Chapter 11 - Rolling, Rolling Torque, and Angular Momentum Rolling Angular Momentum The kinetic energy of the wheel rotating about Conservation of Angular point P is Momentum 1 K = I !2 2 p 1 = I + MR2 !2 2 com 1 1 = I !2 + M(R!)2 2 com 2 1 1 = I !2 + Mv2 2 com 2 K = KR + KT Notice how the force does not oppose the direction of motion! Chapter 11 - Rolling, Rolling Torque, and Angular Momentum Rolling When a rolling object accelerates, the friction Angular Momentum Conservation of Angular force opposes the tendency to slip. Momentum Chapter 11 - Rolling, Rolling Torque, and Angular Momentum Rolling When a rolling object accelerates, the friction Angular Momentum Conservation of Angular force opposes the tendency to slip. Momentum Notice how the force does not oppose the direction of motion! Chapter 11 - Rolling, Rolling Torque, and Angular Momentum Rolling Lecture Question 11.1 Angular Momentum Which one of the following statements concerning a wheel Conservation of Angular Momentum undergoing rolling motion is true? (a) The angular acceleration of the wheel must be 0 m/s2. (b) The tangential velocity is the same for all points on the wheel. (c) The linear velocity for all points on the rim of the wheel is non-zero. (d) The tangential velocity is the same for all points on the rim of the wheel. (e) There is no slipping at the point where the wheel touches the surface on which it is rolling. Why is it defined this way? Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum Rolling An object A with momentum ~p also has angular Angular Momentum Conservation of Angular momentum~l = ~r × ~p about some point O. Momentum Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum Rolling An object A with momentum ~p also has angular Angular Momentum Conservation of Angular momentum~l = ~r × ~p about some point O. Momentum Why is it defined this way? d~r d~p = × ~p +~r × dt dt = ~v × ~p +~r × (m~a) = ~r × ~Fnet = ~τnet d~l τ = net dt Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum From Newton’s 2nd Law, let’s take a derivative of Rolling this “momentum”: Angular Momentum Conservation of Angular Momentum d~l d = (~r × ~p) dt dt = ~v × ~p +~r × (m~a) = ~r × ~Fnet = ~τnet d~l τ = net dt Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum From Newton’s 2nd Law, let’s take a derivative of Rolling this “momentum”: Angular Momentum Conservation of Angular Momentum d~l d = (~r × ~p) dt dt d~r d~p = × ~p +~r × dt dt = ~r × ~Fnet = ~τnet d~l τ = net dt Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum From Newton’s 2nd Law, let’s take a derivative of Rolling this “momentum”: Angular Momentum Conservation of Angular Momentum d~l d = (~r × ~p) dt dt d~r d~p = × ~p +~r × dt dt = ~v × ~p +~r × (m~a) = ~τnet d~l τ = net dt Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum From Newton’s 2nd Law, let’s take a derivative of Rolling this “momentum”: Angular Momentum Conservation of Angular Momentum d~l d = (~r × ~p) dt dt d~r d~p = × ~p +~r × dt dt = ~v × ~p +~r × (m~a) = ~r × ~Fnet d~l τ = net dt Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum From Newton’s 2nd Law, let’s take a derivative of Rolling this “momentum”: Angular Momentum Conservation of Angular Momentum d~l d = (~r × ~p) dt dt d~r d~p = × ~p +~r × dt dt = ~v × ~p +~r × (m~a) = ~r × ~Fnet = ~τnet Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum From Newton’s 2nd Law, let’s take a derivative of Rolling this “momentum”: Angular Momentum Conservation of Angular Momentum d~l d = (~r × ~p) dt dt d~r d~p = × ~p +~r × dt dt = ~v × ~p +~r × (m~a) = ~r × ~Fnet = ~τnet d~l τ = net dt Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum Rolling If there is more than one particle, the total Angular Momentum angular momentum ~L is just the vector sum of the Conservation of Angular Momentum individual angular momentums~li. N X ~L =~l1 +~l2 + ··· +~lN = ~li i=1 And the net torque on the whole system is just: d~L ~τ = net dt Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum The angular momentum of a rigid body can be Rolling found by adding up the angular momentum of the Angular Momentum Conservation of Angular particles that make it up: Momentum li = r?ipi = r?i∆mivi = r?i∆mi(r?i!) 2 = (∆mir?i)! X 2 L = (∆mir?i)! i ! X 2 = ∆mir?i ! i ~L = I~! Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum Rolling Angular Momentum Conservation of Angular Momentum li = r?i∆mivi 2 = (∆mir?i)! X 2 L = (∆mir?i)! i ! X 2 = ∆mir?i ! i ~L = I~! Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum Rolling Angular Momentum Conservation of Angular Momentum li = r?i∆mivi = r?i∆mi(r?i!) X 2 L = (∆mir?i)! i ! X 2 = ∆mir?i ! i ~L = I~! Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum Rolling Angular Momentum Conservation of Angular Momentum li = r?i∆mivi = r?i∆mi(r?i!) 2 = (∆mir?i)! ! X 2 = ∆mir?i ! i ~L = I~! Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum Rolling Angular Momentum Conservation of Angular Momentum li = r?i∆mivi = r?i∆mi(r?i!) 2 = (∆mir?i)! X 2 L = (∆mir?i)! i ~L = I~! Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum Rolling Angular Momentum Conservation of Angular Momentum li = r?i∆mivi = r?i∆mi(r?i!) 2 = (∆mir?i)! X 2 L = (∆mir?i)! i ! X 2 = ∆mir?i ! i Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum Rolling Angular Momentum Conservation of Angular Momentum li = r?i∆mivi = r?i∆mi(r?i!) 2 = (∆mir?i)! X 2 L = (∆mir?i)! i ! X 2 = ∆mir?i ! i ~L = I~! Chapter 11 - Rolling, Angular Momentum Torque, and Angular Momentum Rolling There are many similarities between linear and Angular Momentum Conservation of Angular angular momentum.
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